Note added: Please use scattering times of 10^-13 and 10^-10 instead of the ones suggested in the video.
Benchmark email due asap.
Solved problem due Monday, (May 9)
Problem 1 of your take-home midterm is presented in this video. I would suggesting starting asap and see if you get stuck anywhere. (Unless you are really clear on how to do all this already.) Please post any comments or questions here, and please point out any mistakes, confusing points or lack of adequate specification. Thanks. I look forward to seeing your comments and getting your feedback and emails as you complete the early parts.
This video presents problem 1.
PS. Let's change the scattering times to 10^-10 and 10^-13.
Problem 2 is to calculate the diffusion current just outside the depletion region using the "ideal junction ansatz" with regard to the bifurcation of Ef and its value at the edge of the space-charge region. Calculate the current density as a function of voltage as a function of applied voltage on the right (p side) with the left (n) side grounded. Then also, evaluate that for V_a = 0.6 Volts and A= .01 cm^2.
Additional extra credit: (added Sunday night) At room temperature, what is the thermal velocity of a typical electron near the bottom of the conduction band? Use that to calculate the length scale associated with each of the above scattering times. Discuss those length scales and their relevance to problem 2.
With regard to when it is due, let's say Thursday, and, additionally, your email regarding the numbers you get for n, p, Emax, x_d and V(0) is due Monday by 5 PM. Note that in this case x_d is a little more complicated than before; you can explain your notation in your email.
Extra credit warm-up problem: This will help everybody to have this I think. It is not too hard. Can someone calculate mu = e tau/m in cm^2/Volt-sec . Send me an email and then we will post something soon for everyone to see and use.
Saturday, April 30, 2016
Wednesday, April 27, 2016
Video about np junction I-V characteristics. (Phenomenology, no theory)
Here is a video to watch before class:
Thursday, April 21, 2016
Homework notes
I edited homework 4 to make it shorter and more focused so we can catch up with where we are in class. Please work on that. I think you already understand the issues that HW 3 examines, so let's just skip that if that's okay. Please post comments related to HW 4. Also, I think Ben's post will help a lot with one of the problems (problem 2?). The problems folowing that are related to what we worked on in class.
Monday, April 18, 2016
P-N junction problem from Wednesday
Hello everybody!
Sorry for the delay, but better late than never I suppose.
On Wednesday we were discussing what happens when we place a phosphorous doped semiconductor and a boron doped semiconductor beside one another.
Since the phosphorous atoms in the lattice have an extra valence electron compared to their silicon neighbors, the atoms will want to share electrons with the boron atoms in the lattice of the other semiconductor. The boron atoms having one fewer valence electron in relation to the silicon in the lattice.
Before the semiconductors were connected, the chemical potential \(\mu\) in the n-doped side was at 0.8 eV, and on the p-doped side \(mu=0.2\) eV. We are interested in finding at which value \(\mu\) the system reaches equilibrium in after the semiconductors are connected and electrons flow from n-doped to p-doped.
We started by investigating the field inside the slabs. The n-doped side would be have a negative net charge near the junction and the p-doped side would have a positive net charge. Then, we looked for the field.
If we consider that the distance \(X_{d}\) (called the depletion length) from the junction is ionized in the process, and we assume it carries a uniform charge density \(\rho\), then the field can be determined by Gauss's Law.
The electric field for just one of the slabs is determined as follows
\begin{align*}
\oint\vec{E}\cdot d\vec{a}&=AQ_{enc} \\
2EA&=\rho[AX_{d}]\dfrac{1}{\epsilon_{0}} \\
E&=\dfrac{X_{d}\rho}{2\epsilon_{0}}
\end{align*}
And together, they create a maximum field in the center of junction:
\begin{equation*}
E=X_{d}\dfrac{\rho}{\epsilon_{0}}
\end{equation*}
Since the field is zero at \(x=\pm X_{d}\) and \(X_{d}\dfrac{\rho}{\epsilon_{0}}\) at \(x=0\), and is a linear function of distance:
\begin{equation*}
E=\left\{\begin{array}{r r}
-\dfrac{\rho}{\epsilon_{0}}x+\dfrac{X_{d}\rho}{\epsilon_{0}} & x>0 \\
\dfrac{\rho}{\epsilon_{0}}x+\dfrac{X_{d}\rho}{\epsilon_{0}} & x<0 \\
\end{array}\right.
\end{equation*}
I know we need to find the potential energy of this function next, and also a way to represent \(\rho\) that is workable (density of states maybe). But overall I'm at a little bit of a loss.
Ideas?
edit: fixed a mistake with a factor of two
Saturday, April 16, 2016
Homework 4
Here is a video related to this homework.
For the following problems, consider a semiconductor for which \(E_g = 1 eV, \quad kT=.025 eV\) and
\(D_c = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = 3 eV\)
\(D_v = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_v = 3 eV\).
(Assume that within each band the density of states is independent of E.)
Let's set our zero of energy at the top of the valence band.
Please mention any possible typos, confusing things etc.
It will probably help to discuss a lot of this in the comments here. For example, how do the units work going from charge to electric field to potential? How does one end up with a potential in eV? (You may have to convert from cm to meters when you use surface charge to calculate electric field if epsilon_o is in Farads/meter.)
1. a) If this semiconductor is undoped, then what is the value of n at room temperature? What is the relationship between n and p? What is \(E_f\) for this un-doped case?
b) Suppose that this semiconductor is doped with 10^17 donors/cm^3. In that case we like to assume that each donor contributes one electron to the conduction band. What is \(E_f\) in this case?
(What do you think might be the rationale behind choosing \(12 \times 10^{21} \frac{states}{eV*cm^3}\) for the density of states? Why that value? (post here))
2. a) Plot the density of states as a function of energy from E from E = -4 eV to +4 eV.
b) Calculate n and p for \(E_f = 0.5 eV\).
c) Calculate n and p for \(E_f = 0.2 eV\).
d) Calculate n and p for \(E_f = 0.8 eV\).
e)
Do a semi-log graph of n and p as a function of \(E_f\) for \(E_f\) in
the range 0.1 to 0.9 eV. Can you use the approximate form of the Fermi
function for this calculation?
3. For the same
semiconductor, suppose you dope it with 10^17 donor atoms per cm^3. Then
you get 10^17 electrons/cm^3 in the conduction band and 10^17
positively charged donor atoms/cm^3 embedded in the lattice.
a) What is the charge density associated with just the electrons in the conduction band (in coulombs/cm^3)?
b) What is the charge density associated with the positive ions (in coulombs/cm^3)?
c) What is the net charge density?
d) What would the net charge density be if all the electrons magically disappeared?
2. Consider a semiconductor as above. Suppose that it is doped with 10^17 donors for the half to the left of the plane x=0, and doped with 10^17 acceptors/cm^3 to the right of x=0. (The plane x=0 defines an interface between to two differently doped regions.) Suppose that within a distance \(x_d\) of the interface all the electrons in the conduction band from the left side cross over to the other side and fill up previously empty valance band states (holes) there.
a) When \(E_f\) is independent of x, that represents an equilibrium state. What value of \(x_d\) would enable to bands to bend and shift by just the right amount to enable \(E_f\) to be independent of x?
b) Plot the conduction band edge and valence band edge as a function of x for this case.
6. Suppose the doping on each side is 10^18 cm^-3 instead of 10^17 cm^-3.
a) why might one guess that in that case the equilibrium value of \(x_d\) might be 10 times shorter than for the 10^17 case?
b)
What is the actual equilibrium value of \(x_d\) for this case? Why is
it not exactly 10 times smaller? (What additional factor influences
x_d?)
3 (optional extra problem). Suppose the doping on each side is 10^16 cm^-3.
What is the equilibrium value of \(x_d\) for this case?
4. a) For the n-p junctions you solved in problem 2 (doped with \(10^{17} 1/cm^3\)), use your calculated value for \(E_c (x)\) to calculate n(x) as a function of x. You can use the equation \(n(x) = KT D_c e^{-(E_c (x) - E_f)/kT}\)?
b) What is \(n(-x_d)\)? What is \(n(x_d)\)? Sketch a rough graph n(x) as a function of x.
c) Graph the product of the conduction electron density times the electric fields a function of x. At what x does its maximum value occur?
d) Calculate the drift current as a function of x in the junction region. (You can just do this for x less than zero since that's easier and I think the maximum is in that range.)
5. Use your equation for n(x) from the previous problem to calculate the diffusion current in the junction (for x less than zero). This is an important problem. Please spend some time on this one.
6. Graph this two currents. Compare and discuss here.
Notes: \(n(x) e^2 \tau/m^*m\) multiplied time electric field has units of current per unit area. I think that if you use your expression for n(x) (from the previous problem) and multiply it time the electric field in the junction, which varies linearly with x, that correspond to a current associated with electrons accelerated by the electric field.
Special bonus: Calculate the peak value of that current in Coulombs/second (amperes) for a specific junction with an area of 1 cm^2, using m*=0.2 and \(\tau = 10^{-12}\) seconds. (Use the specific 10^17 doping case above. We want an actual number. Is it big, small, negligible, 10^-15 amps, 2 amps or what?
Hints: One can separate out the term \( \mu= e \tau/m = e \tau c^2/(m^*m c^2)\). With mc^2 in eV and c in cm/s that can have units of \(cm^2/(Volt*seconds\). (The e turns eV into Volts...).
Does the maximum in this product occur somewhere between -x_d to zero? Where is it?
PS. Here is how one might approach graphing current density.
Here is a video on the integration to get V(x):
For the following problems, consider a semiconductor for which \(E_g = 1 eV, \quad kT=.025 eV\) and
\(D_c = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_c = 3 eV\)
\(D_v = 12 \times 10^{21} \frac{states}{eV*cm^3}, \quad B_v = 3 eV\).
(Assume that within each band the density of states is independent of E.)
Let's set our zero of energy at the top of the valence band.
Please mention any possible typos, confusing things etc.
It will probably help to discuss a lot of this in the comments here. For example, how do the units work going from charge to electric field to potential? How does one end up with a potential in eV? (You may have to convert from cm to meters when you use surface charge to calculate electric field if epsilon_o is in Farads/meter.)
1. a) If this semiconductor is undoped, then what is the value of n at room temperature? What is the relationship between n and p? What is \(E_f\) for this un-doped case?
b) Suppose that this semiconductor is doped with 10^17 donors/cm^3. In that case we like to assume that each donor contributes one electron to the conduction band. What is \(E_f\) in this case?
(What do you think might be the rationale behind choosing \(12 \times 10^{21} \frac{states}{eV*cm^3}\) for the density of states? Why that value? (post here))
2. Consider a semiconductor as above. Suppose that it is doped with 10^17 donors for the half to the left of the plane x=0, and doped with 10^17 acceptors/cm^3 to the right of x=0. (The plane x=0 defines an interface between to two differently doped regions.) Suppose that within a distance \(x_d\) of the interface all the electrons in the conduction band from the left side cross over to the other side and fill up previously empty valance band states (holes) there.
a) When \(E_f\) is independent of x, that represents an equilibrium state. What value of \(x_d\) would enable to bands to bend and shift by just the right amount to enable \(E_f\) to be independent of x?
b) Plot the conduction band edge and valence band edge as a function of x for this case.
3 (optional extra problem). Suppose the doping on each side is 10^16 cm^-3.
What is the equilibrium value of \(x_d\) for this case?
4. a) For the n-p junctions you solved in problem 2 (doped with \(10^{17} 1/cm^3\)), use your calculated value for \(E_c (x)\) to calculate n(x) as a function of x. You can use the equation \(n(x) = KT D_c e^{-(E_c (x) - E_f)/kT}\)?
b) What is \(n(-x_d)\)? What is \(n(x_d)\)? Sketch a rough graph n(x) as a function of x.
c) Graph the product of the conduction electron density times the electric fields a function of x. At what x does its maximum value occur?
d) Calculate the drift current as a function of x in the junction region. (You can just do this for x less than zero since that's easier and I think the maximum is in that range.)
5. Use your equation for n(x) from the previous problem to calculate the diffusion current in the junction (for x less than zero). This is an important problem. Please spend some time on this one.
6. Graph this two currents. Compare and discuss here.
Notes: \(n(x) e^2 \tau/m^*m\) multiplied time electric field has units of current per unit area. I think that if you use your expression for n(x) (from the previous problem) and multiply it time the electric field in the junction, which varies linearly with x, that correspond to a current associated with electrons accelerated by the electric field.
Special bonus: Calculate the peak value of that current in Coulombs/second (amperes) for a specific junction with an area of 1 cm^2, using m*=0.2 and \(\tau = 10^{-12}\) seconds. (Use the specific 10^17 doping case above. We want an actual number. Is it big, small, negligible, 10^-15 amps, 2 amps or what?
Hints: One can separate out the term \( \mu= e \tau/m = e \tau c^2/(m^*m c^2)\). With mc^2 in eV and c in cm/s that can have units of \(cm^2/(Volt*seconds\). (The e turns eV into Volts...).
Does the maximum in this product occur somewhere between -x_d to zero? Where is it?
PS. Here is how one might approach graphing current density.
Here is a video on the integration to get V(x):
Homework 3. Not due. Skip this and go to HW 4.
Friday, April 8, 2016
Please comment on HW 2 and start working these problems.
This may be a pretty difficult and long homework. Going from Bloch states to density of states to occupation, and then finally calculating numbers of thermal electrons in the conduction band will probably take some time to sort out. I would suggest starting as soon as you can and spending a couple of sessions working on this and thinking about it over the next few days. Comments and questions are very welcome here. If you are stuck ask questions here, don't wait until class. Also, please reply to other students questions. Please post your results, progress and questions here.
Here is a video discussing how to approach some problems:
--Understanding Bloch states.
1. Using \(\psi_{atom} (x) = \frac{1}{\pi^{1/4}\sqrt{b}}e^{-x^2/2b^2}\) or \(\psi_{atom} (x) = \frac{1}{\sqrt{b}}e^{-|x|/b}\)as your atom state with b= .05 nm:
a) Plot \(\psi_{atom} (x) \) as a function of x.
For each of the next plots, let the range of your plot cover 5 cells, that is, n=-2 to n=+2, and let the crystal lattice parameter, that is, the center to center distance between atoms, be a=.1 nm (= 2b). You can use a computer to learn about these, but your plots must be hand drawn.
b) Plot the Bloch state made from this state for \(k=\pi/a\).
c) Plot the Bloch state made from this state for \(k=\pi/2a\).
d) Plot the Bloch state made from this state for \(k=2\pi/a\).
e) Plot the Bloch state made from this state for \(k= - \pi/2a\).
f) Plot the Bloch state made from this state for \(k=-\pi/a\).
g) Plot the "probability density" for each of the Bloch states above.
h) which of these plots actually require 2 plots and which can be done with only one plot? What is the difference between c) and e)? What is the difference between b) and f)?
--Density of states
2. For a crystal, a total of N energy eigenstates form from each atom energy eigenstate. These states can be kept track of using the crystal quantum number k. Allowed values of k are pretty much \(j 2 \pi/L\) where j is an integer ranging from -N/2 to +N/2 where N is the number of atoms in our 1D crystal and L is the overall length of the crystal (L=Na).
a) Sketch a k axis and draw a small circle each allowed value of k for N=12.
Suppose the energy eigenvalues for each of the states at these discrete values of k are given by the equation \(E_{n,k} = E_n -(B/2) cos(ak)\)
c) Divide the energy axis into 3 equal sections, one from the bottom of the band to B/3 above the bottom; etc. How many states are in each of these 3 sections for N=12?
3. a) For a 1D band, \(E_{n,k} = E_n -(B/2) cos(ak)\), calculate the density of states, D(E), for the limit of large N. You can do that using the idea that D(E) is proportional to the inverse of the derivative of E(k) with respect to k, and the constraint that the integral of D(E) over the band is equal to 2N. You will need to go from k to E after you take the derivative. If that, or anything, is not clear, please ask a question here!
b) Do a graph of D(E) with the vertical scale set for N=100. Use units of eV and assume that B=2 eV.
--working with Density of states
4.1 Sketch the density of states for \(E_2 = -10 eV\) and \(B_2 = 4 eV\), \(E_3 = -4 eV\) and \(B_3 = 6 eV\). Suppose that the E2 band is filled and the E3 band is empty. What is the band gap? How much energy does it take to excite an electron from the top of the filled band to the bottom of the empty band? (email me this number)
4.2 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\): \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many states are occupied if T=0 and \(E_f = -4 eV\).
b) Plot the density of states vs E and show which parts are occupied and which are not.
4.3 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\): \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many states are occupied if T=0 and \(E_f = -7.5 eV\).
b) Plot the density of states vs E and show which parts are occupied and which are not.
c) What is notable about -7.5 eV ?
(email me your a very short description of the situation of 2.3 and 2.4, respectively (ideally just a few words) and your thoughts on the essence of the difference between them.)
4.4 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\): \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many conduction band states are occupied if kT = 25 meV (which corresponds to T= about 295 K or so) and \(E_f = -7.5 eV\).
Actually, that is not a very good way to ask the question. let's ask instead how many thermal electrons there are, on average, in conduction band states.
4.5 Is there an easier way to calculate how many thermal electrons there are, on average, in the conduction band? Approximations you can use in problem 4.4? If so, what are they? How accurate are they in this case?
Here is a video discussing how to approach some problems:
--Understanding Bloch states.
1. Using \(\psi_{atom} (x) = \frac{1}{\pi^{1/4}\sqrt{b}}e^{-x^2/2b^2}\) or \(\psi_{atom} (x) = \frac{1}{\sqrt{b}}e^{-|x|/b}\)as your atom state with b= .05 nm:
a) Plot \(\psi_{atom} (x) \) as a function of x.
For each of the next plots, let the range of your plot cover 5 cells, that is, n=-2 to n=+2, and let the crystal lattice parameter, that is, the center to center distance between atoms, be a=.1 nm (= 2b). You can use a computer to learn about these, but your plots must be hand drawn.
b) Plot the Bloch state made from this state for \(k=\pi/a\).
c) Plot the Bloch state made from this state for \(k=\pi/2a\).
d) Plot the Bloch state made from this state for \(k=2\pi/a\).
e) Plot the Bloch state made from this state for \(k= - \pi/2a\).
f) Plot the Bloch state made from this state for \(k=-\pi/a\).
g) Plot the "probability density" for each of the Bloch states above.
h) which of these plots actually require 2 plots and which can be done with only one plot? What is the difference between c) and e)? What is the difference between b) and f)?
--Density of states
2. For a crystal, a total of N energy eigenstates form from each atom energy eigenstate. These states can be kept track of using the crystal quantum number k. Allowed values of k are pretty much \(j 2 \pi/L\) where j is an integer ranging from -N/2 to +N/2 where N is the number of atoms in our 1D crystal and L is the overall length of the crystal (L=Na).
a) Sketch a k axis and draw a small circle each allowed value of k for N=12.
Suppose the energy eigenvalues for each of the states at these discrete values of k are given by the equation \(E_{n,k} = E_n -(B/2) cos(ak)\)
c) Divide the energy axis into 3 equal sections, one from the bottom of the band to B/3 above the bottom; etc. How many states are in each of these 3 sections for N=12?
3. a) For a 1D band, \(E_{n,k} = E_n -(B/2) cos(ak)\), calculate the density of states, D(E), for the limit of large N. You can do that using the idea that D(E) is proportional to the inverse of the derivative of E(k) with respect to k, and the constraint that the integral of D(E) over the band is equal to 2N. You will need to go from k to E after you take the derivative. If that, or anything, is not clear, please ask a question here!
b) Do a graph of D(E) with the vertical scale set for N=100. Use units of eV and assume that B=2 eV.
--working with Density of states
4.1 Sketch the density of states for \(E_2 = -10 eV\) and \(B_2 = 4 eV\), \(E_3 = -4 eV\) and \(B_3 = 6 eV\). Suppose that the E2 band is filled and the E3 band is empty. What is the band gap? How much energy does it take to excite an electron from the top of the filled band to the bottom of the empty band? (email me this number)
4.2 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\): \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many states are occupied if T=0 and \(E_f = -4 eV\).
b) Plot the density of states vs E and show which parts are occupied and which are not.
4.3 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\): \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many states are occupied if T=0 and \(E_f = -7.5 eV\).
b) Plot the density of states vs E and show which parts are occupied and which are not.
c) What is notable about -7.5 eV ?
(email me your a very short description of the situation of 2.3 and 2.4, respectively (ideally just a few words) and your thoughts on the essence of the difference between them.)
4.4 For the same bands, \(E_2 = -10 eV\) and \(B_2 = 4 eV\): \(E_3 = -4 eV\) and \(B_3 = 6 eV\).
a) How many conduction band states are occupied if kT = 25 meV (which corresponds to T= about 295 K or so) and \(E_f = -7.5 eV\).
Actually, that is not a very good way to ask the question. let's ask instead how many thermal electrons there are, on average, in conduction band states.
4.5 Is there an easier way to calculate how many thermal electrons there are, on average, in the conduction band? Approximations you can use in problem 4.4? If so, what are they? How accurate are they in this case?
Thursday, April 7, 2016
Wednesday, April 6, 2016
100 eV deep square well.
The 100 eV deep finite well
Here is a big final post on this well.
To get the energies we needed to solve \( 1/kb = tan(kL/2) \). To do so I graphed the both sides as functions of E, so \( 1/kb = \sqrt(-\frac{E}{E-V})\) (red graph) and \(tan(kL/2) = tan(L/2 \cdot \frac{\sqrt(2m(E-V))}{\hbar}) \) (green graph).
But this only gives the even states. To get the odd states I chose the odd solution to the schrödinger equation inside the well, Asin(kx), and got
new equation for the energy \( 1/kb = - tan^{-1}(kL/2) \). The new left side is plotted as the pink lines.
So in total I get 4 bound states ( 4 intersections, two even and two odd) with the energies -93, -72, -40, -2.8.
Here is a big final post on this well.
To get the energies we needed to solve \( 1/kb = tan(kL/2) \). To do so I graphed the both sides as functions of E, so \( 1/kb = \sqrt(-\frac{E}{E-V})\) (red graph) and \(tan(kL/2) = tan(L/2 \cdot \frac{\sqrt(2m(E-V))}{\hbar}) \) (green graph).
But this only gives the even states. To get the odd states I chose the odd solution to the schrödinger equation inside the well, Asin(kx), and got
new equation for the energy \( 1/kb = - tan^{-1}(kL/2) \). The new left side is plotted as the pink lines.
Tuesday, April 5, 2016
HW for this week.
This week, please work on on HW1 parts 3 and 4. In part 3 the key problem is number 9, the 4-well problem. Please do really nice drawings of 12 states. Make it clear where zero is on the vertical axis and where the well edges are along the x axis. Use a ruler if you need it. Makes the wells equally spaced and identical in width. Show where the wave-function zero crossings are (nodes). Post lots of questions and comments, not here, but in the HW posts.
In part 4, don't get stuck on derivations or anything like that. Just get the answers anyway you can. The key things are the wave-functions -- being able to visualize, understand and use them. Understanding sp2 and sp3. Understanding degeneracy.
Thanks.
PS. I will work on some band-related HW problems to do next week.
In part 4, don't get stuck on derivations or anything like that. Just get the answers anyway you can. The key things are the wave-functions -- being able to visualize, understand and use them. Understanding sp2 and sp3. Understanding degeneracy.
Thanks.
PS. I will work on some band-related HW problems to do next week.
Friday, April 1, 2016
Videos on delta function and square well wave functions.
These videos explore how to use the wave equation and boundary conditions to find the ground state wave-function for a square well and for an attractive delta function potential.
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